The ACL Learning Framework

The learning framework of ACL differs from ILP because both the background
knowledge and the learned theory
are abductive theories.
An abductive theory T in Abductive Logic Programming is a
triple < P, A, I >, where P is a definite logic program, A is a set
of predicates called
abducible predicates (or simply abducibles),
and I is a set of range-restricted clauses called integrity constraints.As a knowledge representation framework, when we represent a problem
in ALP via an abductive theory T, we generally assume that the abducible
predicates in A carry all the incompleteness of the program P
in modelling the external problem domain in the sense that if we (could)
complete the abducible predicates in P then P
would completely
describe the problem domain. An abductive theory can support abductive
(or hypothetical) reasoning for several purposes such as diagnosis, planning
or default reasoning. The central notion used for this is that of an abductive
explanation for an observation or a given goal. When an observation
O
can be abductively explained in a theory T with explanation Delta
we write
T|=AO. with Delta.
We
can now define the learning problem when the language of the background
and target theories is the one of ALP.

Abductive Concept Learning

Given:

a set of positive examples E+,

a set of negative examples E-,

an abductive theory T=< P, A, I > as background theory,

an hypothesis space T=<H,Y> consisting of a space of possible
programs Hand a space of possible constraints Y

Find:

A set of rules P' belonging to H and a set of constraints
I'
belonging to Ysuch that the new abductive theory T'=< P union
P', A, I union I'> satisfies the following conditions

T'|=AE+,

forall e-belonging to E-,
T'
does
not abductively entail E-.

where E+ stands for the conjunction of all positive examples.

In effect, we have replaced the deductive entailment in the ILP problem
with abductive entailment to define the ACL learning problem.

T' does not abductively entail father(katy,ellen) as the only
possible explanation for this goal, namely {male(katy)} is made inconsistent
by the learned integrity constraints I'

T' does not abductively entail father(john,steve), father(steve,john)
and father(steve,katy) because they have no possible abductive explanations.

Hence, despite the fact that the background theory is incomplete (in its
abducible predicates), ACL can find an appropriate solution to the learning
problem by suitably extending the background theory with abducible assumptions.
Note that the learned theory without the integrity constraint would not
satisfy the definition of ACL, because there would exist an abductive explanation
for the negative example father(katy,ellen), namely {male(katy). This explanation
is prohibited in the complete theory by the learned constraint together
with the fact female(kathy).

The full ACL problem can be split into two subproblems: (1) learning
the rules together with appropriate strong explanations and (2) learning
integrity constraints. The solutions of the two subproblems can be combined
to obtain a solution for the original problem.

The first subproblem, called ACL1, has the following definition.

ACL1

Given:

a set of positive examples E+,

a set of negative examples E-,

an abductive theory T=< P, A, I > as background theory,

an hypothesis space of possible programs H

Find:

A set of rules P' belonging to H such that the new abductive
theory TACL1=< P union P', A, I> satisfies
the following conditions

TACL1|=AE+ with Delta+,

TACL1|=Anot_E- with
Delta-,

Delta+ union Delta- is
consistent

where not_E- stands for the conjunction of the complement
of every negative example.

Example 2

In the case of example 1 above, a solution to
the ACL1 problem would consist of
P'={father(X,Y) :- parent(X,Y),male(X)}
Delta+={male(david)}
Delta-={not_female(katy)}

The ACL1 problem is solved by the system ACL1. The input file for the
learning problem of example 1 is father.bg.
From this input file, the ACL1 system finds the above solution, as can
be seen from the father.rules file.

Indeed, the information generated by ACL1 through the abductive explanations
for negative examples can be used to provide a solution of the full ACL
problem through a second learning phase. From the output of ACL1, i.e.,
its set of rules and the sets of assumptions and for covering positive
examples and uncovering negative ones, a solution to ACL can be found by
learning constraints that are consistent with Delta+and inconsistent
with the complement of every abducible in Delta- .

The definition of the second subproblem, called ACL2, can be given as
follows.

ACL2

Given

a solution of ACL1

TACL1=< P union P', A, I>

Delta+

Delta-

a hypothesis space of possible constraints H

Find

A set of constraints I'ÎYsuch
that the new abductive theory satisfies the following condition

Delta+ is a positive interpretation for
I'

for every literal lbelonging to Delta-,
{not_l} is a negative interpretation
for I'

where not_l is the complement of l with respect to default
negation.

The ACL2 problem is solved by means of the ICL system. The input for
ICL is generated by the ACL1 system, for the case of example 1 the input
files for ICL are again father.bg, containing the
background knowledge, and father.kb, containing
the interpretations, that is generated by the ACL1 system.
The theory, obtained by combining the solutions of the two subproblems,
gives a solution to the full ACL problem.

Example 2

In the case of example 1 and 2
above, the input to the second learning phase, performed by ICL, would
consist of

the positive interpretation p={male(david)}

the negative interpretation n={male(katy)}

The solution of the ACL2 problem is
I'={:- male(X),female(X)}
Indeed the above solution is found by ICL from the above interpretations,
as can be seen from the file father.theory.